The electrical engineering handbook
Dorf, R.C., Wan, Z., Paul, C.R., Cogdell, J.R. “Voltage and Current Sources” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 © 2000 by CRC Press LLC 2 Voltage and Current Sources 2.1 Step, Impulse, Ramp, Sinusoidal, Exponential, and DC Signals Step Function•The Impulse•Ramp Function•Sinusoidal Function•DCSignal 2.2 Ideal and Practical Sources Ideal Sources•Practical Sources 2.3 Controlled Sources What Are Controlled Sources?•What Is the Significance of Controlled Sources?•How Does the Presence of Controlled Sources Affect Circuit Analysis? 2.1 Step, Impulse, Ramp, Sinusoidal, Exponential, and DC Signals Richard C. Dorf and Zhen Wan The important signals for circuits include the step, impulse, ramp, sinusoid, and dc signals. These signals are widely used and are described here in the time domain. All of these signals have a Laplace transform. Step Function The unit-step function u ( t ) is defined mathematically by Here unit step means that the amplitude of u ( t ) is equal to 1 for t ³ 0. Note that we are following the convention that u (0) = 1. From a strict mathematical standpoint, u ( t ) is not defined at t = 0. Nevertheless, we usually take u (0) = 1. If A is an arbitrary nonzero number, Au ( t ) is the step function with amplitude A for t ³ 0. The unit step function is plotted in Fig. 2.1. The Impulse The unit impulse d ( t ), also called the delta function or the Dirac distribution , is defined by ut t t () , , = ³ < ì í ï î ï 10 00 Richard C. Dorf University of California, Davis Zhen Wan University of California, Davis Clayton R. Paul University of Kentucky, Lexington J. R. Cogdell University of Texas at Austin © 2000 by CRC Press LLC The first condition states that d ( t ) is zero for all nonzero values of t , while the second condition states that the area under the impulse is 1, so d ( t ) has unit area. It is important to point out that the value d (0) of d ( t ) at t = 0 is not defined; in particular, d (0) is not equal to infinity. For any real number K , K d ( t ) is the impulse with area K . It is defined by The graphical representation of K d ( t ) is shown in Fig. 2.2. The notation K in the figure refers to the area of the impulse K d ( t ). The unit-step function u ( t ) is equal to the integral of the unit impulse d ( t ); more precisely, we have Conversely, the first derivative of u ( t ), with respect to t , is equal to d ( t ), except at t = 0, where the derivative of u ( t ) is not defined. Ramp Function The unit-ramp function r ( t ) is defined mathematically by Note that for t ³ 0, the slope of r ( t ) is 1. Thus, r ( t ) has unit slope, which is the reason r ( t ) is called the unit-ramp function. If K is an arbitrary nonzero scalar (real num- ber), the ramp function Kr ( t ) has slope K for t ³ 0. The unit-ramp function is plotted in Fig. 2.3. The unit-ramp function r ( t ) is equal to the integral of the unit-step function u ( t ); that is, FIGURE 2.1 Unit-step function. FIGURE 2.2 Graphical representation of the impulse K d ( t ) u ( t ) t 123 1 0 K d( t ) t 0 ( K ) d dll e e e () , () , tt d =¹ = - ò 00 1 for any real number >0 Kt t KdK d dll e e e () , () , =¹ = - ò 00 for any real number >0 ut d t t t () (),= -¥ ò dll all except =0 FIGURE 2.3Unit-ramp function r ( t ) t 123 1 0 rt tt t () , , = ³ < ì í î 0 00 rt u d t () ()= -¥ ò ll © 2000 by CRC Press LLC Conversely, the first derivative of r ( t ) with respect to t is equal to u ( t ), except at t = 0, where the derivative of r(t) is not defined. Sinusoidal Function The sinusoid is a continuous-time signal: A cos(wt + q). Here A is the amplitude, w is the frequency in radians per second (rad/s), and q is the phase in radians. The frequency f in cycles per second, or hertz (Hz), is f = w/2p. The sinusoid is a periodic signal with period 2p/w. The sinusoid is plotted in Fig. 2.4. Decaying Exponential In general, an exponentially decaying quantity (Fig. 2.5) can be expressed as a = A e –t/ t wherea= instantaneous value A= amplitude or maximum value e= base of natural logarithms = 2.718 … t= time constant in seconds t= time in seconds The current of a discharging capacitor can be approxi- mated by a decaying exponential function of time. Time Constant Since the exponential factor only approaches zero as t increases without limit, such functions theoretically last forever. In the same sense, all radioactive disintegrations last forever. In the case of an exponentially decaying current, it is convenient to use the value of time that makes the exponent –1. When t = t = the time constant, the value of the exponential factor is In other words, after a time equal to the time constant, the exponential factor is reduced to approximatly 37% of its initial value. FIGURE 2.4The sinusoid A cos(wt + q) with –p/2 < q < 0. p + 2q 2w p - 2q 2w 3p - 2q 2w 3p + 2q 2w q w A cos(w t + q) 0 – A A t FIGURE 2.5The decaying exponential. ee e t-- === = t 1 11 2718 0368 . . © 2000 by CRC Press LLC DC Signal The direct current signal (dc signal) can be defined mathematically by i(t) = K –¥ < t < +¥ Here, K is any nonzero number. The dc signal remains a constant value of K for any –¥ < t < ¥. The dc signal is plotted in Fig. 2.6. Defining Terms Ramp:A continually growing signal such that its value is zero for t £ 0 and proportional to time t for t > 0. Sinusoid: A periodic signal x(t) = A cos(wt + q) where w = 2pf with frequency in hertz. Unit impulse:A very short pulse such that its value is zero for t ¹ 0 and the integral of the pulse is 1. Unit step:Function of time that is zero for t < t 0 and unity for t > t 0 . At t = t 0 the magnitude changes from zero to one. The unit step is dimensionless. Related Topic 11.1 Introduction References R.C. Dorf, Introduction to Electric Circuits, 3rd ed., New York: Wiley, 1996. R.E. Ziemer, Signals and Systems, 2nd ed., New York: Macmillan, 1989. Further Information IEEE Transactions on Circuits and Systems IEEE Transactions on Education 2.2 Ideal and Practical Sources Clayton R. Paul A mathematical model of an electric circuit contains ideal models of physical circuit elements. Some of these ideal circuit elements (e.g., the resistor, capacitor, inductor, and transformer) were discussed previously. Here we will define and examine both ideal and practical voltage and current sources. The terminal characteristics of these models will be compared to those of actual sources. FIGURE 2.6The dc signal with amplitude K. i ( t ) t 0 K © 2000 by CRC Press LLC Ideal Sources The ideal independent voltage source shown in Fig. 2.7 constrains the terminal voltage across the element to a prescribed function of time, v S (t), as v(t) = v S (t). The polarity of the source is denoted by ± signs within the circle which denotes this as an ideal independent source. Controlled or dependent ideal voltage sources will be discussed in Section 2.3. The current through the element will be determined by the circuit that is attached to the terminals of this source. The ideal independent current source in Fig. 2.8 constrains the terminal current through the element to a prescribed function of time, i S (t), as i(t) = i S (t). The polarity of the source is denoted by an arrow within the FIGURE 2.7Ideal independent voltage source. i(t) + – b a v(t) = v S (t)v S (t) + – v S (t) t A LL -P LASTIC B ATTERY esearchers at the U.S. Air Force’s Rome Laboratory and Johns Hopkins University have developed an all-plastic battery using polymers instead of conventional electrode materials. All-plastic power cells could be a safer, more flexible substitute for use in electronic devices and other commercial applications. In addition, all-polymer cells reduce toxic waste disposal, negate environmental concerns, and can meet EPA and FAA requirements. Applications include powering GPS receivers, communication transceivers, remote sensors, backup power systems, cellular phones, pagers, computing products and other portable equipment. Potential larger applications include remote monitoring stations, highway communication signs and electric vehicles. The Johns Hopkins scientists are among the first to create a potentially practical battery in which both of the electrodes and the electrolyte are made of polymers. Fluoro-substituted thiophenes polymers have been developed with potential differences of up to 2.9 volts, and with potential specific energy densities of 30 to 75 watt hours/kg. All plastic batteries can be recharged hundreds of times and operate under extreme hot and cold temperature conditions without serious performance degradation. The finished cell can be as thin as a business card and malleable, allowing battery manufacturers to cut a cell to a specific space or make the battery the actual case of the device to be powered. (Reprinted with permission from NASA Tech Briefs, 20(10), 26, 1996.) R © 2000 by CRC Press LLC circle which also denotes this as an ideal independent source. The voltage across the element will be determined by the circuit that is attached to the terminals of this source. Numerous functional forms are useful in describing the source variation with time. These were discussed in Section 2.1—the step, impulse, ramp, sinusoidal, and dc functions. For example, an ideal independent dc voltage source is described by v S (t) = V S , where V S is a constant. An ideal independent sinusoidal current source is described by i S (t) = I S sin(wt + f) or i S (t) = I S cos(wt + f), where I S is a constant, w = 2pf with f the frequency in hertz and f is a phase angle. Ideal sources may be used to model actual sources such as temperature transducers, phonograph cartridges, and electric power generators. Thus usually the time form of the output cannot generally be described with a simple, basic function such as dc, sinusoidal, ramp, step, or impulse waveforms. We often, however, represent the more complicated waveforms as a linear combination of more basic functions. Practical Sources The preceding ideal independent sources constrain the terminal voltage or current to a known function of time independent of the circuit that may be placed across its terminals. Practical sources, such as batteries, have their terminal voltage (current) dependent upon the terminal current (voltage) caused by the circuit attached to the source terminals. A simple example of this is an automobile storage battery. The battery’s terminal voltage is approximately 12 V when no load is connected across its terminals. When the battery is applied across the terminals of the starter by activating the ignition switch, a large current is drawn from its terminals. During starting, its terminal voltage drops as illustrated in Fig. 2.9(a). How shall we construct a circuit model using the ideal elements discussed thus far to model this nonideal behavior? A model is shown in Fig. 2.9(b) and consists of the series connection of an ideal resistor, R S , and an ideal independent voltage source, V S = 12 V. To determine the terminal voltage–current relation, we sum Kirchhoff’s voltage law around the loop to give (2.1) This equation is plotted in Fig. 2.9(b) and approximates that of the actual battery. The equation gives a straight line with slope –R S that intersects the v axis (i = 0) at v = V S . The resistance R S is said to be the internal resistance of this nonideal source model. It is a fictitious resistance but the model nevertheless gives an equivalent terminal behavior. Although we have derived an approximate model of an actual source, another equivalent form may be obtained. This alternative form is shown in Fig. 2.9(c) and consists of the parallel combination of an ideal independent current source, I S = V S /R S , and the same resistance, R S , used in the previous model. Although it may seem strange to model an automobile battery using a current source, the model is completely equivalent to the series voltage source–resistor model of Fig. 2.9(b) at the output terminals a–b. This is shown by writing Kirchhoff’s current law at the upper node to give FIGURE 2.8Ideal independent current source. v(t) + – b a i(t) = i S (t) i S (t) i S (t) t vVRi SS =- © 2000 by CRC Press LLC (2.2) Rewriting this equation gives (2.3) Comparing Eq. (2.3) to Eq. (2.1) shows that (2.4) FIGURE 2.9Practical sources. (a) Terminal v-i characteristic; (b) approximation by a voltage source; (c) approximation by a current source. i v 12V i v V S = 12V Slope = –R S i v V S = 12V Slope = –R S I S Automobile Storage Battery + – + – v b a i V S + + – v b a i – R S V S + – v b a i R S R S I S = (a) (b) (c) iI R v S S =- 1 vRIRi SS S =- VRI SSS = © 2000 by CRC Press LLC Therefore, we can convert from one form (voltage source in series with a resistor) to another form (current source in parallel with a resistor) very simply. An ideal voltage source is represented by the model of Fig. 2.9(b) with R S = 0. An actual battery therefore provides a close approximation of an ideal voltage source since the source resistance R S is usually quite small. An ideal current source is represented by the model of Fig. 2.9(c) with R S = ¥. This is very closely represented by the bipolar junction transistor (BJT). Related Topic 1.1 Resistors Defining Term Ideal source:An ideal model of an actual source that assumes that the parameters of the source, such as its magnitude, are independent of other circuit variables. Reference C.R. Paul, Analysis of Linear Circuits, New York: McGraw-Hill, 1989. 2.3 Controlled Sources J. R. Cogdell When the analysis of electronic (nonreciprocal) circuits became important in circuit theory, controlled sources were added to the family of circuit elements. Table 2.1 shows the four types of controlled sources. In this section, we will address the questions: What are controlled sources? Why are controlled sources important? How do controlled sources affect methods of circuit analysis? What Are Controlled Sources? By source we mean a voltage or current source in the usual sense. By controlled we mean that the strength of such a source is controlled by some circuit variable(s) elsewhere in the circuit. Figure 2.10 illustrates a simple circuit containing an (independent) current source, i s , two resistors, and a controlled voltage source, whose magnitude is controlled by the current i 1 . Thus, i 1 determines two voltages in the circuit, the voltage across R 1 via Ohm’s law and the controlled voltage source via some unspecified effect. A controlled source may be controlled by more than one circuit variable, but we will discuss those having a single controlling variable since multiple controlling variables require no new ideas. Similarly, we will deal only with resistive elements, since inductors and capacitors introduce no new concepts. The controlled voltage or current source may depend on the controlling variable in a linear or nonlinear manner. When the relationship is nonlinear, however, the equations are frequently linearized to examine the effects of small variations about some dc values. When we linearize, we will use the customary notation of small letters to represent general and time-variable voltages and currents and large letters to represent constants such as the dc value or the peak value of a sinusoid. On subscripts, large letters represent the total voltage or current and small letters represent the small-signal component. Thus, the equation i B = I B + I b cos wt means that the total base current is the sum of a constant and a small-signal component, which is sinusoidal with an amplitude of I b . To introduce the context and use of controlled sources we will consider a circuit model for the bipolar junction transistor (BJT). In Fig. 2.11 we show the standard symbol for an npn BJT with base (B), emitter (E), and collector (C) identified, and voltage and current variables defined. We have shown the common emitter configuration, with the emitter terminal shared to make input and output terminals. The base current, i B , ideally depends upon the base-emitter voltage, v BE , by the relationship © 2000 by CRC Press LLC (2.5) where I 0 and V T are constants. We note that the base current depends on the base-emitter voltage only, but in a nonlinear manner. We can represent this current by a voltage-controlled current source, but the more common representation would be that of a nonlinear conductance, G BE (v BE ), where Let us model the effects of small changes in the base current. If the changes are small, the nonlinear nature of the conductance can be ignored and the circuit model becomes a linear conductance (or resis- tor). Mathematically this conductance arises from a first-order expan- sion of the nonlinear function. Thus, if v BE = V BE + v be , where v BE is the total base-emitter voltage, V BE is a (large) constant voltage and v be is a (small) variation in the base-emitter voltage, then the first two terms in a Taylor series expansion are TABLE 2.1Names, Circuit Symbols, and Definitions for the Four Possible Types of Controlled Sources Name Circuit Symbol Definition and Units Current-controlled voltage source (CCVS) v 2 = r m i 1 r m = transresistance units, ohms Current-controlled current source (CCCS) i 2 = bi 1 b, current gain, dimensionless Voltage-controlled voltage source (VCVS) v 2 = mv 1 m, voltage gain, dimensionless Voltage-controlled current source (VCCS) i 2 = g m v 1 g m , transconductance units, Siemans (mhos) r m v 2 + – + – i 1 i 1 bi 1 i 1 i 2 v 2 + – + – mv 1 v 1 + – g m v 1 + – i 2 v 1 FIGURE 2.10A simple circuit con- taining a controlled source. FIGURE 2.11An npn BJT in the com- mon emitter configuration. iI v V B BE T = é ë ê ù û ú ì í î ï ü ý þ ï 0 1exp – Gv i v BE BE B BE ()=